N. T. Vo, F. Winkler

@inproceedings{RISC5194, author = {N. T. Vo and F. Winkler}, title = {{Algebraic General Solutions of First Order Algebraic ODEs}}, booktitle = {{Computer Algebra in Scientific Computing}}, language = {english}, abstract = {In this paper we consider the class of algebraic ordinary differential equations (AODEs), the class of planar rational systems, and discuss their algebraic general solutions. We establish for each parametrizable first order AODE a planar rational system, the associated system, such that one can compute algebraic general solutions of the one from the other and vice versa. For the class of planar rational systems, an algorithm for computing their explicit algebraic general solutions with a given rational first integral is presented. Finally an algorithm for determining an algebraic general solution of degree less than a given positive integer of parametrizable first order AODEs is proposed.}, series = {Lecture Notes in Computer Science}, volume = {9301}, pages = {479--492}, publisher = {Springer International Publishing}, isbn_issn = {ISSN 0302-9743}, year = {2015}, editor = {Vladimir P. Gerdt et. al.}, refereed = {yes}, length = {14}, url = {http://link.springer.com/content/pdf/10.1007%2F978-3-319-24021-3_35.pdf}}

2014

The Concept of Gröbner Reduction for Dimension in filtered free modules

@techreport{RISC5068, author = {Christoph Fuerst and Guenter Landsmann}, title = {{The Concept of Gröbner Reduction for Dimension in filtered free modules}}, language = {english}, abstract = {We present the concept of Gröbner reduction that is a Gröbner basistechnique on filtered free modules. It allows to compute the dimensionof a filtered free module viewn as a K-vector space. We apply the de-veloped technique to the computation of a generalization of Hilbert-typedimension polynomials in K[X] as well as in finitely generated difference-differential modules. The latter allows us to determine a multivariatedimension polynomial where we partition the set of derivations and theset of automorphism in a difference-differential ring in an arbitrary way.}, number = {14-12}, year = {2014}, month = {October}, length = {13}, type = {RISC Report Series},institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},address = {Schloss Hagenberg, 4232 Hagenberg, Austria}}

Towards a Theory of Discrete and Mesoscopic Wave Turbulence

E.Kartashova, V. Lvov, S. Nazarenko, I. Procaccia

@techreport{RISC3967, author = {E.Kartashova and V. Lvov and S. Nazarenko and I. Procaccia}, title = {{Towards a Theory of Discrete and Mesoscopic Wave Turbulence}}, language = {english}, abstract = {This is WORK IN PROGRESS carried out in years 2008-2009 and partly supported by Austrian FWF-project P20164-N18 and 6 EU Programme under the project SCIEnce, Contract No. 026133).Abstract:\emph{Discrete wave turbulence} in bounded media refers to the regular and chaotic dynamics of independent (that is, discrete in $k$-space) resonance clusters consisting of finite (often fairly big) number of connected wave triads or quarters, with exact three- or four-wave resonances correspondingly. "Discreteness" means that for small enough amplitudes there is no energy flow among the clusters. Increasing of wave amplitudes and/or of system size opens new channels of wave interactions via quasi-resonant clusters. This changes statistics of energy exchange between waves and results in new, \emph{mesoscopic} regime of \emph{wave turbulence}, where \emph{discrete wave turbulence} and \emph{kinetic wave turbulence} in unbounded media co-exist, the latter well studied in the framework of wave kinetic equations. We overview in systematic manner and from unified viewpoint some preliminary results of studies of regular and stochastic wave behavior in bounded media, aiming to shed light on their relationships and to clarify their role and place in the structure of a future theory of discrete and mesoscopic wave turbulence, elucidated in this paper. We also formulate a set of yet open questions and problems in this new field of nonlinear wave physics, that awaits for comprehensive studies in the framework of the theory. We hope that the resulting theory will offer very interesting issues both from the physical and the methodological viewpoints, with possible important applications in environmental sciences, fluid dynamics, astronomy and plasma physics.}, year = {2010}, month = {February}, howpublished = {Technical report no. 10-04 in RISC Report Series}, length = {42}, type = {RISC Report Series},institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},address = {Schloss Hagenberg, 4232 Hagenberg, Austria}}

Turbulence of capillary waves revisited

E. Kartashova, A. Kartashov

@article{RISC4026, author = {E. Kartashova and A. Kartashov}, title = {{Turbulence of capillary waves revisited}}, language = {english}, abstract = {Kinetic regime of capillary wave turbulence is classically regarded in terms of three-wave interactions with the exponent of power energy spectrum being $\nu=-7/4$ (two-dimensional case). We show that a number of assumptions necessary for this regime to occur can not be fulfilled. Four-wave interactions of capillary waves should be taken into account instead, which leads to exponents $\nu=-13/6$ and $\nu=-3/2$ for one- and two-dimensional wavevectors correspondingly. It follows that for general dispersion functions of decay type, three-wave kinetic regime need not prevail and higher order resonances may play a major role.}, journal = {EPL}, volume = {submitted}, pages = {1--6}, isbn_issn = {isbn}, year = {2010}, refereed = {yes}, length = {6}, url = {http://arxiv.org/abs/1005.2067}}